The proposed research will develop and study statistical methods for analyzing key properties of infectious disease epidemics when data on times of infection and disease are inexact. The techniques developed and knowledge gained will enable epidemiologists to better monitor changing infection rates, quantify rates of progression to disease, and identify and quantify risk factors for infection and for rapid disease progression. Study of these aspects of an epidemic is important for projecting its future impact, planning and evaluating preventive interventions, and targetting therapy and providing appropriate care. The methods apply to studies where infection times are interval-censored because subjects are periodically tested for evidence of inapparent infections, where many subjects are already infected at the time of recruitment, and where disease times determine which subjects are available for study. Techniques for analyzing overall population patterns of diagnoses over time will also be studied. These types of incomplete data have been extensively analyzed for the epidemic of HIV and AIDS, and such data are commonly collected for other infectious disease epidemics, as well. For all the methods, algorithms will be implemented to optimize roughness-penalized likelihood criteria, thereby fitting nonparametric models that balance fit to the data with the need for plausibly smooth estimates. Techniques will be applied to real data sets, thereby addressing epidemiologic questions that are of interest in their own right. The performance of the techniques will be studied by also applying them to simulated data sets where the true models are known. Variations and refinements of all techniques will be implemented and compared. Additive modelling and partial likelihood methods will be used to estimate the influence of covariates on infection rates and times between infection and disease, and the use of cross-validation and other methods for choosing models of optimal complexity will be studied. Bootstrap resampling and simulation methods will be developed for estimating pointwise confidence intervals and bias corrections. All methods will be compared to competing imputation, parametric, and unsmoothed nonparametric methods using both real and simulated data.